Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry

Transcription

1 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry Daniel Huybrechts Institut de Mathématiques Jussieu, Université Paris 7, France Lectures given at the School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 LNS

2 Abstract These lectures treat some of the basic features of moduli spaces of hyperkähler manifolds and in particular of K3 surfaces. The relation between the classical moduli spaces and the moduli spaces of conformal field theories is explained from a purely mathematical point of view. Recent results on hyperkähler manifolds are interpreted in this context. The second goal is to give a detailed account of mirror symmetry of K3 surfaces. The general principle, due to Aspinwall and Morrison, and various special cases (e.g. mirror symmetry for lattice polarized or elliptic K3 surfaces) are discussed.

3 Contents 1 Introduction Basics Moduli spaces Moduli spaces of marked IHS Moduli spaces of marked HK Moduli spaces of marked complex HK or Kähler IHS CFT moduli spaces of HK Moduli spaces without markings Period domains Positive (oriented) subspaces Planes and complex lines Planes and three-spaces Three- and four-spaces Pairs of planes Calculations in the Mukai lattice Topology of period domains Density results Period maps Definition of the period maps Geometry and period maps The diffeomorphism group of a K3 surface (Derived) Global Torelli Theorem Discrete group actions Maximal discrete subgroups Geometric symmetries Mirror symmetries The mirror map U U Geometric interpretation of mirror symmetry Lattice polarized mirror symmetry Mirror symmetry by hyperkähler rotation Mirror symmetry for elliptic K3 surfaces

4 7.4 FM transformation and mirror symmetry Large complex structure limit References 244

5 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry Introduction Let (M, g) be a compact Riemannian manifold of dimension 2N with holonomy group SU(N). If N > 2 then there is a unique complex structure I on M such that g is a Kähler metric with respect to I. For given g and I a symplectic structure on M is naturally defined by the associated Kähler form ω = g(i( ), ). This construction locally around g yields a decomposition of the moduli space of all Calabi-Yau metrics on M as M met (M) = M cpl (M) M khl (M), where M cpl (M) is the moduli space of complex structures on M and M khl (M) is the moduli space of symplectic structures. Mirror symmetry in a first approximation predicts for any Calabi-Yau manifold (M, g) the existence of another Calabi-Yau manifold (M v, g v ) together with an isomorphism M met (M) = M met (M v ) which interchanges the two factors of the above decomposition, e.g. M cpl (M) = M khl (M v ). The picture has to be modified when we consider the second type of irreducible Ricci-flat manifolds. If the holonomy group of a 4n-dimensional manifold (M, g) is Sp(n), i.e. (M, g) is a hyperkähler manifold, then the moduli space of metrics near g does not split into the product of complex and kähler moduli as above, e.g. for a given hyperkähler metric there is a whole sphere S 2 of complex structures compatible with g. Hence mirror symmetry as formulated above for Calabi-Yau manifolds needs to be reformulated for hyperkähler manifolds. It still defines an isomorphism between the metric moduli spaces, but the relation between complex and symplectic structures are more subtle. Nevertheless, mirror symmetry is supposed to be much simpler for hyperkähler manifolds, as usually the mirror manifold M v as a real manifold is M itself. These notes intend to explain the analogue of the product decomposition of the moduli space of metrics on a Calabi-Yau manifold in the hyperkähler situation and to show how mirror symmetry for K3 surfaces, i.e. hyperkähler manifolds of dimension 4, is obtained by the action of a discrete group. After recalling the main definitions and facts concerning the complex and metric structure of these manifolds in Section 2 we will soon turn to the global aspects of their moduli spaces. In Sections 3 and 4 we introduce these moduli spaces as well as the corresponding period domains. The geometric moduli spaces are studied via maps into the period domains. This will be explained in Section 5. Some of the main results about compact hyperkähler manifolds can be translated into global aspects of these maps.

6 190 D. Huybrechts Compared to other texts (e.g. [1]) on moduli spaces of K3 surfaces we will try to develop the theory as far as possible for compact hyperkähler manifolds of arbitrary dimension. The second main difference is that we also treat the less classical moduli spaces of certain CFTs. This will be done from a purely mathematical point of view by considering hyperkähler manifolds endowed with an additional B-field, i.e. a real cohomology class of degree two. This will lead to new features starting in Section 6, where we let act a certain discrete group on the various moduli spaces. This section follows papers by Aspinwall, Morrison, and others. Using this action mirror symmetry of K3 surfaces will be explained in Section 7. The advantage of this slightly technical approach is that various versions of mirror symmetry for (e.g. lattice polarized or elliptic) K3 surfaces can be explained by the same group action. Of course, explaining mirror symmetry in these terms is only possible for K3 surfaces or hyperkähler manifolds. Mirror symmetry for general Calabi Yau manifolds will usually change the topology. The text contains little or no original material. The main goal was to explain global phenomena of moduli spaces of K3 surfaces, or more generally of compact hyperkähler manifolds, and to give a concise introduction into the main constructions used in establishing mirror symmetry for K3 surfaces. We encourage the reader to consult the survey [1] and the original articles [3, 4]. 2 Basics In this section we collect the basic definitions and facts concerning irreducible holomorphic symplectic manifolds and compact hyperkähler manifolds. Most of the material will be presented without proofs and we shall refer to other sources for more details (e.g. [6, 20]). Definition 2.1 An irreducible holomorphic symplectic manifold (IHS, for short) is a simply connected compact Kähler manifold X, such that H 0 (X, Ω 2 X ) is generated by an everywhere non-degenerate holomorphic two-form σ. Since an IHS is in particular a compact Kähler manifold, Hodge decomposition holds. In degree two it yields H 2 (X, C) = H 2,0 (X) H 1,1 (X) H 0,2 (X) = Cσ H 1,1 (X) C σ.

7 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 191 The existence of an everywhere non-degenerate two-form σ H 0 (X, Ω 2 X ) implies that the manifold has even complex dimension dim C (X) = 2n. Moreover, σ induces an alternating homomorphism σ : T X Ω X. Since the twoform is everywhere non-degenerate, this homomorphism is bijective. Thus, the tangent bundle and the cotangent bundle of an IHS are isomorphic. Moreover, the canonical bundle K X = Ω 2n X is trivialized by the (2n, 0)-form σ n. Thus, an IHS has trivial canonical bundle and, therefore, vanishing first Chern class c 1 (X). In dimension two IHS are also called K3 surfaces (K3=Kähler, Kodaira, Kummer). More precisely, by definition a K3 surface is a compact complex surface with trivial canonical bundle K X and such that H 1 (X, O X ) = 0. It is a deep fact that any such surface is also Kähler [40]. Moreover, H 1 (X, O X ) = 0 does indeed imply that such a surface is simply-connected. Here are the basic examples. Examples 2.2 i) Any smooth quartic hypersurface X P 3 is a K3 surface, e.g. the Fermat quartic defined by x x4 1 + x4 2 + x4 3 = 0. ii) Let T = C 2 / be a compact two-dimensional complex torus. The involution x x has 16 fixed points and, thus, the quotient T/± is singular in precisely 16 points. Blowing-up those yields a Kummer surface X T/±, which is a K3 surface containing 16 smooth irreducible rational curves. iii) An elliptic K3 surface is a K3 surface X together with a surjective morphism π : X P 1. The general fibre of π is a smooth elliptic curve. It is much harder to construct higher dimensional examples of IHS and all known examples are constructed by means of K3 surfaces or two-dimensional complex tori. The list of known examples has been discussed in length in the lectures of Lehn (see also [20]). So far we have discussed IHS purely from the complex geometric point of view. However, the most important feature of this type of manifolds is the existence of a very special metric. Definition 2.3 A compact oriented Riemannian manifold (M, g) of dimension 4n is called hyperkähler (HK, for short) if the holonomy group of g equals Sp(n). In this case g is called a hyperkähler metric. Remark 2.4 If g is a hyperkähler metric, then there exist three complex structures I, J, and K on M, such that g is Kähler with respect to all three

8 192 D. Huybrechts of them and such that K = I J = J I. Thus, I is orthogonal with respect to g and the Kähler form ω I := g(i( ), ) is closed (similarly for J and K). Often, this is taken as a definition of a hyperkähler metric. Note that our condition is stronger, as we not only want the holonomy be contained in Sp(n), but be equal to it. Proposition 2.5 Let (M, g) be a HK. Then for any (a, b, c) R 3 with a 2 + b 2 + c 2 = 1 the complex manifold (M, ai + bj + ck) is an IHS. Thus, for any HK (M, g) there exists a two-sphere S 2 R 3 of complex structures compatible with the Riemannian metric g. Remark 2.6 Let (M, g) be a HK. The associated Kähler forms ω I, ω J, ω K span a three-dimensional subspace H 2 + (M, g) H2 (M, R). In fact, this space will always be considered as a three-dimensional space endowed with the natural orientation. If X = (M, I), then H 2 +(M, g) = (H 2,0 (X) H 0,2 (X)) R Rω I, where the orientation is given by the base (Re(σ), Im(σ), ω I ). In order to see this, one verifies that the holomorphic two-form σ on X = (M, I) can be given as σ = ω J + iω K (cf. [20]). Definition 2.7 Let X be an IHS. The Kähler cone K X H 1,1 (X, R) is the open convex cone of all Kähler classes on X, i.e. classes that can be represented by some Kähler form. The most important single result on IHS is the following consequence of the celebrated theorem of Calabi Yau: Theorem 2.8 Let X be an IHS. Then for any α K X there exists a unique hyperkähler metric g on M, such that α = [ω I ] for ω I = g(i( ), ). Thus, on any IHS X the Kähler cone K X parametrizes all possible hyperkähler metrics g compatible with the given complex structure. Below we will explain how the Kähler cone K X can be described as a subset of H 1,1 (X). Remark 2.9 Thus, an IHS X together with a Kähler class α K X is the same thing as a HK (M, g) together with a compatible complex structure I. As a short hand, we write (X, α) = (M, g, I) in this case.

9 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 193 Definition 2.10 The BB(Beauville Bogomolov)-form of an IHS X is the quadratic form on H 2 (X, R) given by q X (α) = n α 2 (σ σ) n 1 + (1 n)( ασ n 1 σ n )( ασ n σ n 1 ), 2 X X where σ H 2,0 (X) is chosen such that X (σ σ)n = 1 For any Kähler class [ω] we obtain a q X -orthogonal decomposition H 2 (X, R) = (H 2,0 (X) H 0,2 (X)) R Rω H 1,1 (X) ω. Here, H 1,1 (X) ω is the space of ω- primitive real (1, 1)-classes. Note that we get a different decomposition for every Kähler class [ω] K X, but that the quadratic form q X does not depend on the chosen Kähler class. The following proposition collects the main facts about the BB-form q X. Proposition 2.11 i) For any Kähler class [ω] K X on an IHS X the BBform q X is positive definite on (H 2,0 (X) H 0,2 (X)) R Rω and negative definite on H 1,1 (X) ω. ii) There exists a positive real scalar λ 1 such that q X (α) n = λ 1 X α2n for all α H 2 (X). iii) There exists a positive real scalar λ 2 such that λ 2 q X is a primitive integral form on H 2 (X, Z). iv) There exists a positive real scalar λ 3 such that q X (α) = λ 3 X α2 td(x) for all α H 2 (X). After eliminating the denominator of td(x) by multiplying with a universal coefficient c n that only depends on n we obtain an integral quadratic form c n α 2 td(x). In general this form need not be primitive, but this will be of no importance for us. Moreover, since any IHS has vanishing odd Chern classes, td(x) = Â(X). (Everything that matters here is that td(x) is purely topological in this case.) Therefore, in these lectures we will use the following modified version of the BB-form. Definition 2.12 The BB-form q X of an 2n-dimensional IHS X is given by q X (α) = c n α Â(X). 2 With this definition we see that q X only depends on the underlying manifold M, i.e. for two different hyperkähler metrics g and g and two compatible complex structures I resp. I the BB-forms with the above definition of X = (M, I) and X = (M, I ) coincide. X X

10 194 D. Huybrechts Note for n = 1 we have c 1 = 1 and thus q X is nothing but the intersection pairing α α of the four-manifold underlying a K3 surface. The quadratic form in this case is even, unimodular and indefinite and can thus be explicitly determined: Proposition 2.13 The intersection form (H 2 (X, Z), ) of a K3 surface X is isomorphic to the K3 lattice 2( E 8 ) 3U, where U is the standard hyperbolic plane ( Z 2, ( )). Definition 2.14 The BB-volume of a HK (M, g) is q(m, g) := q X ([ω I ]), where X = (M, I) is the IHS associated to one of the compatible complex structures I and ω I is the induced Kähler form. Note that the BB-volume does not depend on the chosen complex structure. Analogously one can define the volume of an IHS endowed with a Kähler class α as q X (α). For a K3 surface one has q(m, g) = ωi 2, which is the usual volume up to the scalar factor 1/2. In higher dimension the usual volume is of degree 2n and the BB-volume is quadratic. Of course, due to Proposition 2.11 one knows that up to a scalar factor q(m, g) n equals the standard volume, but this factor might a priori depend on the topology of M. What makes the theory of K3 surfaces and higher-dimensional HK so pleasant is that they can be studied by means of their period. Definition 2.15 Let X be an IHS. The period of X is the lattice (H 2 (X, Z), q X ) endowed with the weight-two Hodge structure H 2 (X, Z) C = H 2 (X, C) = Cσ H 1,1 (X, C) C σ. Since H 1,1 (X, C) is orthogonal with respect to q X and C σ is the complex conjugate of Cσ, the period of the IHS X is in fact given by the lattice (H 2 (X, Z), q X ) and the line Cσ H 2 (X, C). The theory of K3 surfaces is crowned by the so called Global Torelli Theorem (due to Pjateckii-Sapiro, Shafarevich, Burns, Rapoport, Looijenga, Peters, Friedman): Theorem 2.16 Let X and X be two K3 surfaces and let ϕ : H 2 (X, Z) = H 2 (X, Z) be an isomorphism of their periods such that ϕ(k X ) K X. Then there exists a unique isomorphism f : X = X such that f = ϕ.

11 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 195 Moreover, an arbitrary isomorphism between the periods of two K3 surfaces is in general not induced by an isomorphism of the K3 surfaces, but the K3 surfaces are nevertheless isomorphic. The uniqueness assertion in the Global Torelli Theorem is roughly proven as follows (cf. [29]): If f is a non-trivial automorphism of finite order with f = id then the holomorphic two-form σ is invariant under f and the action at the fixed points is locally of the form (u, v) (ξ u, ξ 1 v). Using Lefschetz fixed point formula and again f = id one finds that there are 24 fixed points. Thus, the minimal resolution X of the quotient X/ f contains 24 pairwise disjoint curves. Moreover, one verifies that X is again a K3 surface. The last two statements together yield a contradiction. The Global Torelli Theorem in the above version fails completely in higher dimensions. E.g. if f : X = X is an automorphism of a K3 surface X such that f = id, then f = id. This does not hold in higher dimensions [7]. Even worse, due to a recent counterexample of Namikawa [36] one knows that higher dimensional IHS X and X might have isomorphic periods without even being birational. A possible formulation of the Global Torelli Theorem in higher dimensions using derived catgeories was proposed in [36]. However, uniqueness is not expected. Compare the discussion in Section 5.4. Often, a certain type of K3 surfaces is distinguished by the form of the period. We explain this in the three examples presented earlier. In fact, the proofs of these descriptions are all quite involved. Example 2.17 i) Let X be a K3 surface such that Pic(X) = H 2 (X, Z) H 1,1 (X) is generated by a class α with α 2 = 4. Then X is isomorphic to a quartic hypersurface in P 3 and α corresponds to O(1) (cf. [1, Exp. VI]). ii) Let X be a K3 surface such that Pic(X) contains 16 disjoint smooth irreducible rational curves C 1,..., C 16 X such that [C i ] H 2 (X, Z) is two-divisible. Then X is isomorphic to a Kummer surface. This description of Kummer surfaces is not entirely in terms of the period. Later we will rather use the following description of an even more special type of K3 surfaces: Let X be a K3 surface such that the lattice (H 2,0 (X) H 0,2 (X)) Z is of rank two and any vector x in this lattice satisfies x 2 0 mod 4. Then X is a Kummer surface. It turns out that K3 surfaces with this type of period are exactly the exceptional Kummer surfaces, i.e. Kummer surfaces with rk(pic(x)) = 20 (cf. [1, Exp.VIII]).

12 196 D. Huybrechts iii) Let X be a K3 surface such that there exists a class α H 2 (X, Z) H 1,1 (X) with α 2 = 0. Then X is an elliptic K3 surface. Clearly, if X P 1 is an elliptic K3 surface then the class of the fibre defines such a class. But note that conversely not every class α with α 2 = 0 is automatically a fibre class of some elliptic fibration, but by applying certain reflections it can be be made into one (cf. [8]). In order to get a better feeling for the set of all possible hyperkähler structures on an IHS X we shall discuss the Kähler cone in some more detail. Definition 2.18 The positive cone C X of an IHS X is the connected component of the open set {α q X (α) > 0} H 1,1 (X, R) that contains the Kähler cone K X. (Here we use the fact that q X (α) > 0 for any Kähler class α.) Thus, C X ( C X ) can be entirely read off the period of X. This is no longer possible for the Kähler cone, but one can at least try to find a minimal set of further geometric information that determines K X as an open subcone of C X. Proposition 2.19 The Kähler cone K X C X is the open subset of all α C X such that C α > 0 for all rational curves C X. If X is a K3 surface it suffices to test smooth rational curves (cf. [1, 5, 20]). Since any smooth irreducible rational curve C in a K3 surface X defines a ( 2)-class [C] H 1,1 (X, Z), one can use this result to show that for any class α C X there exists a finite number of smooth rational curves C 1,..., C k X such that s C1... s Ck (α) K X, where s C is the reflection in the hyperplane [C]. Of course, these reflections s C are contained in the discrete orthogonal group O() of the lattice = (H 2 (X, Z), ). 3 Moduli spaces Ultimately, we will be interested in moduli spaces of irreducible holomorphic symplectic manifolds (IHS), hyperkähler manifolds (HK), etc. In this section we will introduce moduli spaces of such manifolds endowed with an additional marking. A marking in general refers to an isomorphism of the second cohomology with a fixed lattice. The choice of such an isomorphism

13 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 197 gives rise to the action of a discrete group and the quotients by this group will eventually yield the true moduli spaces. For this section we fix a lattice of signature (3, b 3) and an integer n. 3.1 Moduli spaces of marked IHS Definition 3.1 A marked IHS is a pair (X, ϕ) consisting of an IHS of complex dimension 2n and a lattice isomorphism ϕ : (H 2 (X, Z), q X ) =. We say that two marked IHS (X, ϕ) and (X, ϕ ) are equivalent, (X, ϕ) (X, ϕ ), if there exists an isomorphism f : X = X of complex manifolds such that ϕ = ϕ f. Definition 3.2 The moduli space of marked IHS is the space T cpl := {(X, ϕ) = marked IHS}/. A priori, T cpl is just a set, but, as we will see later, it can be endowed with the structure of a topological space locally isomorphic to a complex manifold of dimension b 2. Let X be an IHS and ϕ a marking of X. If X Def(X) is the universal deformation of X = X 0, then Def(X) is a smooth germ of dimension h 1 (X, T X ). We may represent Def(X) by a small disc in C h1 (X,T X ). The marking ϕ induces in a canonical way a marking ϕ t of the fibre X t for any t Def(X). Using the Local Torelli Theorem (cf. Section 5) we see that the induced map Def(X) T cpl is injective, i.e. any two fibres of the family X Def(X) define non-equivalent marked IHS. The various Def(X) T cpl for all possible choices of X and markings ϕ cover the moduli space T cpl. Since the universal deformation X Def(X) of X = X 0 is, at the same time, also the universal deformation of all its fibres X t, one can define a natural topology on T cpl by gluing the complex manifolds Def(X). Thus, locally T cpl is a smooth complex manifold of dimension h 1 (X, T X ) = b 2. However, T cpl is not a complex manifold, as it does not need to be Hausdorff. In fact, no example is known, where T cpl would be Hausdorff and conjecturally this never happens. A family (X, ϕ) S of marked IHS is a family X S of IHS of dimension 2n and a family of markings ϕ t of the fibres X t locally constant with respect to t. The universality of X Def(X) immediately implies the following

14 198 D. Huybrechts Lemma 3.3 If (X, ϕ) S is a family of marked IHS, then there exists a canonical holomorphic map η : S T cpl, such that η(t) = [(X t, ϕ t )]. Remark 3.4 In order to construct a universal family over T cpl one would need to glue universal families X Def(X), Y Def(Y ), where (X, ϕ) and (Y, ψ) are marked IHS, over the intersection Def(X) Def(Y ) T cpl. This is only possible if for t Def(X) Def(Y ) there exists a unique isomorphism f : X t = Yt with ψ t = ϕ t f. For K3 surfaces the uniqueness can be ensured due to the strong version of the Global Torelli Theorem (see Thm. 2.16), but in higher dimensions this fails. Thus, T cpl is, in general, only a coarse moduli space. 3.2 Moduli spaces of marked HK Definition 3.5 A marked HK is a triple (M, g, ϕ), where (M, g) is a compact HK of dimension 4n in the sense of Proposition 2.3 and ϕ is an isomorphism (H 2 (M, Z), q) =. Two triples (M, g, ϕ), (M, g, ϕ ) are equivalent, (M, g, ϕ) (M, g, ϕ ), if there exists an isometry f : (M, g) = (M, g ) with ϕ = ϕ f. Definition 3.6 The moduli space of marked HK is the space T met := {(M, g, ϕ) = marked HK}/. A slightly different approach towards T met will be explained in Section 3.5. There, the manifold M is fixed and only the metric g is allowed to vary. 3.3 Moduli spaces of marked complex HK or Kähler IHS Recall (cf. Remark 2.9) that there is a bijection between HK with a compatible complex structure and IHS with a chosen Kähler class. Thus, the two moduli spaces are naturally equivalent. Definition 3.7 A marked complex HK is a tuple (M, g, I, ϕ), where (M, g, ϕ) is a marked HK and I is a compatible complex structure on (M, g). A marked Kähler IHS is a triple (X, α, ϕ), where (X, ϕ) is a marked IHS and α K X is a Kähler class. Two marked complex HK (M, g, I, ϕ), (M, g, I, ϕ ) are equivalent if there exists an isometry f : (M, g) = (M, g ) with I = f I and ϕ = ϕ f. Analogously, one defines the equivalence of marked Kähler IHS.

15 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 199 Note that the equivalence relation is compatible with the natural bijection {(M, g, I, ϕ)} {(X, α, ϕ)}. Definition 3.8 The moduli space of complex HK or, equivalently, of Kähler IHS is the space T := {(M, g, I, ϕ) = marked complex HK}/ = {(X, α, ϕ) = marked Kähler IHS}/. Obviously, there are two forgetful maps m : (M, g, I, ϕ) (M, g, ϕ) and c : (X, α, ϕ) (X, ϕ). The following diagram is the hyperkähler version of the product decomposition of the metric moduli space for Calabi Yau manifolds. T m c T cpl T met Proposition 3.9 The set T has the structure of a real manifold of dimension 3(b 2). The fibre c 1 (X, ϕ) = K X is a real manifold of dimension b 2. The fibre m 1 (M, g, ϕ) is naturally isomorphic to the complex manifold P 1. The induced map c : P 1 = m 1 (M, g, ϕ) T cpl is a holomorphic embedding. The map m : c 1 (X, ϕ) T met is a real embedding. The line P 1 T cpl is also called twistor line. Disposing of a global deformation like this, is one of the key tools in studying moduli spaces of IHS. 3.4 CFT moduli spaces of HK From a geometric point of view the following moduli space is an almost trivial extension of T. However, it will become of central interest in later sections, when we will let act the full modular group on it. This group action will relate very different HK and thus gives rise to mirror symmetry phenomena. Definition 3.10 A marked complex HK with a B-field is a tuple (M, g, I, B, ϕ), where (M, g, I, ϕ) is a marked complex HK and B H 2 (M, R). Two such tuples (M, g, I, B, ϕ), (M, g, I, B, ϕ ) are equivalent if there exists an isometry f : (M, g) = (M, g ) with I = f I, ϕ = ϕ f, and f (B ) = B.

16 200 D. Huybrechts Definition 3.11 The (2, 2)-CFT moduli space of HK is the space T (2,2) := {(M, g, I, B, ϕ) = marked complex HK with B field}/. Clearly, the moduli space T (2,2) is naturally isomorphic to T R by mapping (M, g, I, B, ϕ) to ((M, g, I, ϕ), ϕ R (B)). In particular, T (2,2) is a real manifold of dimension 4b 6. Analogously, one defines the (4, 4)-CFT moduli space T (4,4) := {(M, g, B, ϕ) = marked HK with B field}/. which is sur- In particular, there is a natural forgetful map T (2,2) T (4,4) jective with fibre S Moduli spaces without markings All previous moduli spaces parametrize various geometric objects with an additional marking of the second cohomology. Of course, what we are really interested in are the true moduli spaces M cpl, Mmet, M, M (2,2), and M (4,4). E.g. M cpl is the moduli space of IHS X of dimension 2n such that (H 2 (X, Z), q X ) is isomorphic to, but without actually fixing the isomorphism. Analogously for the other spaces. In other words one has: M cpl = O() \ T cpl, M met = O() \ T met, M = O() \ T, M (2,2) = O() \ T (2,2), M (4,4) = O() \ T (4,4). The Teichmüller spaces T are in general better behaved. E.g. the moduli spaces are usually singular at points that correspond to manifolds with a bigger automorphism group than expected. This usually leads to orbifold singularities. However, sometimes the passage from the Teichmüller space to the moduli space is really ill-behaved. E.g. the action of O() on T cpl is not properly discontinuous. Thus, T cpl which already is not Hausdorff, becomes even worse when divided out by O() (cf. the discussion in Section 6). There is yet another approach to these moduli spaces where one actually fixes the underlying manifold and constructs the moduli space as a quotient of the space of hyperkähler metrics by the diffeomorphism group. We will briefly discuss this.

17 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 201 Let M be a compact oriented differentiable manifold of real dimension 4n and let q M be the quadratic form on H 2 (M, Z) given by q M (α) = c n M Â(M). α2 We write = (H 2 (M, Z), q M ) and call this identification ϕ 0. By Diff(M) we denote the group of orientation-preserving diffeomorphisms of M. In fact, at least for b 2 6, the group Diff(M) is the full diffeomorphism group of M, as any orientation-reversing diffeomorphism f would induce an isomorphism (H 2 (M, Z), q M ) = (H 2 (M, Z), q M ) which is impossible for b 2 (M) 6. The set of all hyperkähler metrics g on M is denoted by Met HK (M). Clearly, Diff(M) acts naturally on Met HK (M) by (f, g) f g. Definition 3.12 The group Diff o (M) Diff(M) is the connected component of Diff(M) containing the identity id M Diff(M). The group Diff (M) Diff(M)) is the kernel of the natural representation Diff(M) O(H 2 (M, Z), q M ). Mapping g Met HK (M) to (M, g, ϕ 0 ) T met diagram induces a commutative Met HK (M)/Diff (M) Met HK (M)/Diff(M) η η T met M met Note that η is well-defined. Indeed, if f Diff (M), then (M, g, ϕ 0 ) (M, f g, ϕ 0 f ) = (M, f g, ϕ 0 ). Remark 3.13 It seems essentially nothing is known about the quotient of the natural inclusion Diff o (M) Diff (M), not even for K3 surfaces, i.e. n = 1. T met Clearly, the image of η (and η) can contain only those HK (M, g, ϕ) whose underlying real manifold M is diffeomorphic to M. Let T met (M) (M) denote the union of all those connected components. and M met i) In general, η : Met HK (M)/Diff (M) T met (M) is injective, but not surjective. The injectivity is clear. Let us explain why surjectivity fails in general. If (M, g, ϕ) Im(η) and ψ O(H 2 (M, Z), q M ), then (M, g, ϕ 0 ψ) Im(η) if and only if there exists f Diff(M) with f = ψ but Diff(M) O(H 2 (M, Z), q M ) is not necessarily surjective. E.g. for K3 surfaces the image does not contain id and, more precisely, O(H 2 (M, Z), )/Diff(M) = Z/2Z

18 202 D. Huybrechts (cf. 5.3). However in this case the situation is rather simple, as T met consists of two components, interchanged by id H 2, and Met HK (M)/Diff (M) is one of them. For higher dimensional HK nothing is known about the image of Diff(M) O(H 2 (M, Z), q M ). ii) The map η : Met HK (M)/Diff(M) M met (M) is bijective. Indeed, if (M, g, ϕ) T met (M), then [(M, g, ϕ 0 )] = [(M, g, (ϕ 0 ϕ 1 )ϕ)] = (ϕ 0 ϕ 1 )[(M, g, ϕ)] = [(M, g, ϕ)] M met (M). Thus, η is surjective. If η(m, g) = η(m, g ), then there exists ψ O() such that (M, g, ϕ 0 ) (M, g, ψ ϕ 0 ) and hence there exists f Diff(M) with f g = g (note that for b 2 (M) = 6 one would have to argue that f can be chosen orientationpreserving) and ϕ 0 = ψ ϕ 0 f. Thus, [(M, g)] = [(M, g )] in Met HK (M)/Diff(M) and hence η is injective. One last word concerning the stabilizer of the action of Diff(M). Clearly, the stabilizer of a hyperkähler metric g is the isometry group Isom(M, g) of (M, g). This group is compact (cf. [9]). Hence the stabilizer of g Met HK (M) is a compact group. Moreover, Isom(M, g) Diff (M) is finite. Indeed, if f Isom(M, g) then f maps any g-compatible complex structure I to another g-compatible complex structure f I. If in addition f = id on H (M, R), then the map I f I must also be the identity. Hence f Aut(M, I) for any g-compatible complex structure I. Since H 0 ((M, I), T ) = 0, the latter group is discrete and, therefore, Aut(M, I) Isom(M, g) is finite. Hence, the action of Diff (M) on Met HK (M) has finite stabilizer. 4 Period domains The moduli spaces that have been introduced in the last section will be studied by means of various period maps. In this section we define and discuss the spaces in which these maps take their values, the period domains. Let be a lattice of signature (m, n). The standard example for is the K3 lattice 2( E 8 ) 3U, where U denotes the hyperbolic plane ( Z 2, ( )) However, might in general be non-unimodular. This will be of no importance in this section, as only the real vector space R := R is going to be used. In fact, usually we will work with an arbitrary vector space V, but will nevertheless occur in the notation. I hope this will not lead to any confusion.

19 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry Positive (oriented) subspaces Let V be a real vector space that is endowed with a bilinear form, of signature (m, n), e.g. V = R. We will also write x 2 for x, x. Fix k m and consider the space of all k-dimensional subspaces W V such that, restricted to W is positive definite. We will denote this space by Gr p k (V ). Clearly, Gr p k (V ) is an open non-empty subset of the Grassmannian Gr k(v ). In order to describe Gr p k (V ) as a homogeneous space we consider the natural action of O(V ) on Gr p k (V ) given by (ϕ, W ) ϕ(w ). The stabilizer of a point W 0 Gr p k (V ) is O(W 0) O(W0 ). Since the action is transitive, one obtains the following description Gr p k (V ) = O(V )/ O(W0 ) O(W0 ) = O(m, n)/ O(k) O(m k, n) The second isomorphism depends on the choice of a basis of the spaces W 0 and W0. Next consider the space k (V ) of all oriented positive subspaces W V of dimension k. Clearly, the natural map k (V ) Grp k (V ) is a 2 : 1 cover. Again, O(V ) acts transitively on k (V ) and the stabilizer of an oriented positive subspace W 0 is SO(W 0 ) O(W0 ). Thus, 2 (V ) = O(V )/ SO(W0 ) O(W 0 ) = O(m, n)/ SO(k) O(m k, n) 4.2 Planes and complex lines For k = 2 the space 2 (V ) allows an alternative description. It turns out that there is a natural bijection between this space and the space Q := {x x 2 = 0, (x + x) 2 > 0} P( C ), where we use the C-linear extension of,. Note that the second condition in the definition of Q is well posed, i.e. independent of the representative x C of the line x P( C ), as long as the first condition x 2 = 0 is satisfied. Clearly, Q is an open subset of a non-singular quadric hypersurface in P( C ). To any x Q one associates the plane W x := R (xc xc) R endowed with the orientation given by (Re(x), Im(x)). Since xc xc is invariant under conjugation, this space is indeed a real plane. Moreover, W λx =

20 204 D. Huybrechts R (λxc λ xc) = W x and (Re(λx), Im(λx)) = (Re(x), Im(x)) ( Re(Λ) Im(λ) ) Im(λ) Re(Λ), where the matrix has positive determinant. Hence, the oriented plane W x is well-defined, i.e. it only depends on x P( C ). It is positive, since (λx + λ x) 2 = λ λ(x + x) 2 > 0 for λ 0. Conversely, if W 2 ( R), then choose a positively oriented orthonormal basis w 1, w 2 W and set x := w 1 + iw 2. Then W = W x and x 2 = 0, (x + x) 2 = (2w 1 ) 2 > 0. Moreover, x P( C ) does not depend on the choice of the basis and any x Q can be written in this form. Thus, one has a bijection 4.3 Planes and three-spaces Q = Gr po 2 ( R) For our purpose the spaces 2 ( R), 3 ( R), and 4 ( R U R ) are the most interesting ones. In the next two sections we will study how they are related to each other. To this end let us first introduce the space 2,1 ( R) := {(P, ω) P 2 ( R), ω P R, ω 2 > 0}. Clearly, this space projects naturally to 2 ( R) by (P, ω) P. The fibre over the point P is the quadratic cone {ω ω 2 > 0} P R. If has signature (3, b 3), this cone consists of exactly two connected components, which can be identified with each other by ω ω. Thus, the fibre of 2,1 ( R) 2 ( R) over P in this case is the disjoint union of two copies of a connected cone, which will be called C P. In fact, 2,1 ( R) 2 ( R) is a trivial cover, i.e. 2,1 ( R) splits into two components. This can either be deduced from the fact that 2 ( R) = O(3, b 3)/(SO(2) O(1, b 3)) is simply connected (cf. Section 4.7) or from the following argument: If we fix an oriented positive three-space F 3 ( R), then the orthogonal projection P Rω F for any ω ±C X must be an isomorphism, since F is negative. Thus, we can single out one of the two connected components of ±C P by requiring that P Rω = F is compatible with the orientations on both spaces. Mapping (P, ω) to the oriented positive three-space F (P, ω) := P ωr and the scalar ω 2 R >0 defines a map 2,1 ( R) 3 ( R) R >0. The map is surjective and the fibre over a point (F, λ) can be identified with the set of all ω F with ω 2 = λ which is a two-dimensional sphere.

21 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 205 Thus, one has the following diagram 2,1 ( R) S 2 3 ( R) R >0 = ( O(m, n) / SO(3) O(m 3, n) ) R >0 ±C P 2 ( R) = O(m, n)/ SO(2) O(m 2, n) Note that the two natural compositions S 2 2,1 ( R) 2 ( R) and C P 2,1 ( R) Gr p 3 ( R) R >0 are both injective. 4.4 Three- and four-spaces From now on we will assume that has signature (3, b 3). Furthermore, let us fix a standard basis (w, w ) of U, i.e. w 2 = w 2 = 0 and w, w = 1. We will see that the space of four-spaces in R U R relates naturally to the space of three-spaces in R. Explicitly, we will show 3 ( R) R >0 R = Gr po 4 ( R U R ) = O(4, b 2)/ SO(4) O(b 2) The second isomorphism follows from Section 4.1. The first one is given as follows. φ : (F, α, B) Π := B R F, where F := {f f, B w f F } and B := B (α B2 )w + w. Clearly, f f, B w, B = f f, B w, B + w = 0 and thus the decomposition is orthogonal. Furthermore, (f f, B w) 2 = f 2 > 0 for 0 f F and B 2 = B 2 +α B 2 = α > 0. Hence, Π is a positive four-space. Its orientation is induced by the orientation of F = F and the decomposition Π = B R F. In order to see that φ is bijective we study the inverse map ψ : Π (F, B 2, B), where F, B, and B are defined as follows: One first introduces F := Π w. This space is of dimension three, since otherwise Π w = R wr and the latter space does not contain any positive four-space. Again by the positivity of Π one finds w F Π. Hence, F := π(f ) R is a positive three-space, where π : R U R R is the natural projection. Furthermore, there exists a B Π such that Π = B R F is an orthogonal

22 206 D. Huybrechts splitting. As before B cannot be contained in w. Thus, one can rescale B such that B, w = 1. This determines B uniquely. Since B Π, one has B 2 > 0. The B-field is by definition B := π(b ). One easily verifies that ψ and φ are indeed inverse to each other. 4.5 Pairs of planes The last space we will discuss in this series of period domains is the space of orthogonal oriented positive planes in R U R, i.e. 2,2 ( R U R ) = {(H 1, H 2 ) H i 2 ( R U R ), H 1 H 2 }. Using the same techniques as before this space can also be described as an homogeneous space as follows 2,2 ( R U R ) = O( R U R )/ SO(H1 ) SO(H 2 ) O((H 1 H 2 ) ) = O(4, b 2)/ SO(2) SO(2) O(b 2), for some chosen point (H 1, H 2 ) 2,2 ( R U R ), respectively basis of the spaces H 1, H 2, and (H 1 H 2 ). We will be interested in the natural projection π : 2,2 ( R U R ) 4 ( R U R ), (H 1, H 2 ) Π := H 1 H 2 and in the injection γ : 2,1 ( R) R 2,2 ( R U R ) which is compatible with 2,1 ( R) 3 ( R). Let us first study the projection. Using the above description of both spaces as homogeneous spaces this map corresponds to dividing by SO(4)/(SO(2) SO(2)). The fibre of π over Π 4 ( R U R ) is canonically isomorphic to 2 (Π) via (H 1, H 2 ) H 1. The inverse image of H 2 (Π) is (H, H ), where H gets its orientation from Π and the decomposition Π = H H. Thus, one obtains the following description of the fibre π 1 (Π) = 2 (Π) = S 2 S 2.

23 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 207 The second isomorphism is derived as in Section 4.2 from 2 (Π) = {x P(Π C) x 2 = 0} = P 1 P 1. Note that (x + x) 2 > 0 is automatically satisfied, for, on Π is positive by assumption. Let us now turn to the injection γ, which is defined as follows. We set γ((p, ω), B) = (H 1, H 2 ) with and H 2 := H 1 := {x x, B w x P } ( ) 1 2 (α B2 )w + w + B R (ω ω, B w) R, where as before (w, w ) is the standard basis of U and α = ω 2. The isomorphism P = H 1, x x x, B w endows H 1 with an orientation. A natural orientation of H 2 is given by definition. Observe that H 1 only depends on P and B, whereas H 2 depends on ω and B. One easily verifies that the map γ is injective and that it commutes with the projections to 3 ( R U R ) R >0 R = Gr po 4 ( R U R ). Recall that the fibre of 2,1 ( R) R 4 ( R U R ) is S 2, whereas the fibre of π : 2,2 ( R U R ) 4 ( R U R ) is S 2 S 2. It can be checked that the embedding γ does not identify the fibre S 2 neither with the diagonal nor with one of the two factors. In algebro-geometric terms S 2 S 2 S 2 is a hyperplane section of P 1 P 1 with respect to the Segre embedding in P 3. Remark 4.1 Note that the projection 2,1 ( R) R 2,1 ( R) 2 ( R) does not extend, at least not canonically, to a map 2,2 ( R U R ) 2 ( R). Geometrically this will be interpreted by the fact that not any point in the (2, 2)-CFT moduli space of K3 surfaces canonically defines a complex structure. More recently, it has become clear that generalized K3 surfaces, a notion that relies on Hitchin s generalized Calabi-Yau structures [22], might be useful to give a geometric interpretation to every N = (2, 2)-SCFT (see [27]) We summarize the discussion of this paragraph in the following commutative diagram

24 208 D. Huybrechts 2,2 ( R U R ) S 2 S 2 4 ( R U R ) 2,1 ( R) R S 2 3 ( R) R >0 R R 2,1 ( R) S 2 R 3 ( R) R >0 ±C P 2 ( R) 4.6 Calculations in the Mukai lattice We shall indicate how the formulae change if we pass to the Mukai bilinear form. This will enable us to make the description of the various period spaces and period maps compatible with conventions used elsewhere. We include this discussion for completeness, but it is not necessary for the understanding of the later sections. Clearly the hyperbolic lattice U with the standard basis w, w is isomorphic to U via w w, w w. This extends to a lattice isomorphism η : U = ( U) =:. For any oriented four-manifold M underlying a K3 surfaces we can identify H (M, Z) endowed with the standard intersection pairing with U such that w = 1 H 0 (M, Z), w = [pt] H 4 (M, Z), and = H 2 (M, Z). Then is naturally isomorphic to H (M, Z) with the Mukai pairing (α 0 + α 2 + α 4, β 0 + β 2 + β 4 ) = α 0 β 4 α 4 β 0 + α 2 β 2, where α i, β i H i (M, Z). The identification of U and with the cohomology of a K3 surface induces a ring structure on both lattices, i.e. in both cases we define (λw + x + µw) 2 := λ 2 w + 2λx + (2λµ + x 2 )w. Note that η does not respect these ring structures. Using the ring structure on R we can let act any element B 0 R on R via its exponential exp(b 0 ) = w + B 0 + (B 2 0 /2)w. Lemma 4.2 For any B 0 R one has exp(b 0 ) O( R ).

25 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 209 Proof. This results from the following straightforward calculation = (exp(b 0 ) (λw + x + µw)) 2 ( λw + (λb 0 + x) + (µ + B 0, x + λ B2 0 2 )w = x 2 2λµ = (λw + x + µw) 2. Later we shall study the map ϕ B0 associated to any B 0 R (see Section 6.2. By definition ϕ B0 O( R U R ) acts on R U R by w w, w B 0 + w B2 0 2 w, x x B 0, x w, for x R. Let us compare exp(b 0 ) with ϕ B0, Proposition 4.3 Under the isomorphism η : R U R = R the automorphism ϕ B0 corresponds to the action of exp(b 0 ), i.e. η ϕ B0 = exp(b 0 ) η. Proof. By definition, exp(b 0 ) acts by ) 2 w w, w (w + B 0 + B2 0 2 w) w = B 0 + w + B2 0 2 w, x (w + B 0 + B2 0 2 w) x = x + B 0, x w, for x R, which yields the assertion. The isomorphism η induces a natural isomorphism 2,2 ( R U R ) = 2,2 ( R ). In order, to describe the image η(h 1, H 2 ) we will use the identification Q = Gr po 2 ( R ) established in Section 4.2. The positive plane associated to an element x C with [x] Q will be denoted by P x, i.e. P x is spanned by Re(x) and Im(x). Clearly, exp(b) P x = P exp(b)x. Let (P, ω) 2,1 ( R) and B R. We denote (H 1, H 2 ) := γ((p, ω), 0) and (H1 B, HB 2 ) := γ((p, ω), B). Then a direct calculation shows ϕ B(H 1, H 2 ) = (H1 B, HB 2 ) and therefore Corollary 1 η(h B 1, HB 2 ) = exp(b) η(h 1, H 2 ) = exp(b) (P σ, P exp(iω) ) = (P exp(b)σ, P exp(b+iω) ).

26 210 D. Huybrechts Proof. The only thing that needs a proof is η(h 2 ) = P exp(iω). But this follow immediately from the definition of H 2. In particular, via η the image of γ : 2,1 ( R) R 2,2 ( R U R ) ( ) can be identified with exp( R ) η(γ( 2,1 ( R))). 4.7 Topology of period domains Let us study some basic aspects of the topology of the period domains that are of interest for us. Let be a lattice of signature (3, b 3). We will consider the spaces: 2 ( R) = O(3, b 3)/ SO(2) O(1, b 3) 4 ( R U R ) = O(4, b 2)/ SO(4) O(b 2) 3 ( R) = O(3, b 3)/ SO(3) O(b 3) 2,2 ( R U R ) = O(4, b 2)/ SO(2) SO(2) O(b 2) For simplicity we will suppose that b > 3. Lemma 4.4 The group O(k, l) with k, l > 0 has exactly four connected components. Proof. Write O := O(k, l). Then there are the following disjoint unions O = O + O, O = O + O, and O = O + + O+ O + O. Here, O± ± are defined as follows: Write R k+l = W 0 W0 with W 0 R k+l a maximal positive subspace, which is endowed with an orientation. Then let O + and O (respectively, O + and O ) be the subsets of all linear maps A O such that the orthogonal projection AW 0 W 0 (respectively, AW0 W0 ) is orientation preserving resp. orientation reversing. By definition O + + = O + O +, etc. For any A 0 O ± ± the map O+ + O± ±, A AA 0 defines a homeomorphism. Thus, it suffices to show that O + + is connected. Note that O + + (k, l) is the connected component of the identity. It will thus also be denoted O o (m, n). Corollary 2 The space 2 ( R) is connected, whereas the spaces 2,1 ( R), 3 ( R), 4 ( R U R ), and 2,2 ( R U R ) consist of two connected components.

27 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 211 Proof. Use the obvious fact that the inclusion SO(2) O(1, b 3) O(3, b 3) respects the decomposition into connected components, i.e. O ± ± (1, b 3) O ± ± (3, b 3). Thus, π 0(SO(2) O(1, b 3)) = π 0 (O(3, b 3)). Similarly for 3 ( R). Here O(3, b 3) has four connected components, but π 0 (SO(3) O(b 3)) = Z/2Z, i.e. the components O ± do not intersect the image of the inclusion. Hence, π 0 ( 3 ( R)) = Z/2Z. The remaining assertions are proved analogously. We are also interested in the fundamental groups of these spaces. order to compute those, we recall the following classical facts. Proposition 4.5 One has π 1 (SO(2)) = Z, π 1 (SO(k)) = Z/2Z for k > 2, and π 1 (O o (k, l)) = π 1 (SO(k)) π 1 (SO(l)). Proof. The first assertion follows from SO(2) = S 1. The universal cover of SO(k) for k 3 is the two-to-one cover Spin(k) SO(k). The isomorphism in the last assertion is induced by the natural inclusion SO(k) SO(l) O o (k, l). Corollary 3 All the Grassmanians 2 ( R), 2,1 ( R), 3 ( R), 4 ( R U R ), and 2,2 ( R U R ) are simply-connected, i.e. every connected component is simply connected. Proof. Since 2 ( R) = O(3, b 3)/ SO(2) O(1, b 3) = O o (3, b 3)/ SO(2) Oo (1, b 3), we may use the exact sequence π 1 (SO(2) O o (1, b 3)) a π 1 (O o (3, b 3)) π 1 ( 2 ( R)) π 0 ( ) = π 0 ( ). The map a is compatible with the natural isomorphisms π 1 (SO(2) O o (1, b 3)) = π 1 (SO(2)) π 1 (O o (1, b 3)) = π 1 (SO(2)) π 1 (SO(1)) π 1 (SO(b 3)), π 1 (O o (3, b 3)) = π 1 (SO(3)) π 1 (SO(b 3)) and the natural maps Z = π 1 (SO(2)) π 1 (SO(1)) π 1 (SO(3)) = Z/2Z. Thus, a is surjective and hence π 1 ( 2 ( R)) = 0. The other assertions are proved analogously. Remark 4.6 Eventually, we list the real dimensions of our period spaces, which can easily be computed starting from 2 ( R) = Q. We have dim 2 ( R) = 2(b 2), dim 2,1 ( R) = 3(b 2), dim 3 ( R) = 3(b 3), 4 ( R U R ) = 4(b 2), and dim 2,2 ( R U R ) = 4(b 1). In

28 212 D. Huybrechts 4.8 Density results Here we shall be interested in those points P Q whose orthogonal complement P R contains integral elements α of given length. For simplicity we shall assume that is the K3 lattice 2( E 8 ) 3U, but all we will use is that is even of index (3, b 3) and that any primitive isotropic element of can be complemented to a sublattice of which is isomorphic to the hyperbolic plane. First note the following easy fact. Lemma 4.7 If 0 α R then α Q is not empty. Proof. Indeed, α R is a hyperplane containing at least two linearly independent orthogonal positive vectors x, y. Thus, P := x, y α Q. The quadric in P( C ) defined by the quadratic form, on will be denoted Z, its real points form the set Z R = P( R ) Z. Proposition 4.8 Let 0 α. Then the set g(α Q ) = g(α) Q is dense in Q. g O() g O() Proof. We start out with the following observation: Let = U be an orthogonal decomposition and let (v, v ) be a standard basis of U. For B with B 2 0 we define ϕ B O() by ϕ B (v) = v, ϕ B (v ) = B +v B 2 /2 v, and ϕ B (x) = x B, x v for x. It is easy to see that indeed with this definition ϕ B O(). (We shall study a similarly defined automorphism ϕ B O( U) in Section 6). This automorphism has the remarkable property that for any y R one has lim k ϕk N[y] = [v] P( R ). In particular, we find that in the closure of the orbit O := O() [α] P( R ) there exists an isotropic vector, i.e. O Z R. In order to prove the assertion of the proposition we have to show that for any P Q there exists an automorphism g O() such that g(α) is arbitrarily close to P. Indeed, in this case we find a codimension two subspace W R close to P containing g(α) and, therefore, W Q is close to P and orthogonal to g(α).

29 Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry 213 Since P contains some isotropic vector, it suffices to show that any vector [y] Z R P( R ) is contained in O. As explained before, O Z R. On the other hand, O Z R is closed and O()-invariant. Thus, it suffices to show that any O()-orbit O y := O() [y] Z R is dense. This is proved in two steps. i) The closure O y contains the subset {[x] Z x }. Indeed, for any x primitive with x 2 = 0 one finds an orthogonal decomposition = U with x = v, where (v, v ) is a standard basis of the hyperbolic plane U. If we choose B with B 2 0, then lim k ϕ k B [y] = [v] = [x], as we have seen before. Hence, [x] O y. ii) The set {[x] Z x } is dense in Z. Indeed, if we write = U as before, then the dense open subset V Z R of points of the form [x + λv + v ] with λ R, x R is the affine quadric {(x, λ) 2λ + x 2 = 0} R R and thus is given as the graph of the rational polynomial R R, x x 2 /2. Therefore, the rational points are dense in V. Combining both steps yields the assertion. Corollary 4 For any m Z the subset {P Q there exists a primitive α P with α 2 = 2m} is dense in Q. Proof. In order to apply the proposition we only have to ensure that there is a primitive element 0 α with α 2 = 2m. If (w, w ) is the standard base of a copy of the hyperbolic plane U contained in, we can choose α = w + mw. In fact, if α 1, α 2 are primitive elements with α 2 1 = α2 2 then there exists an automorphism ϕ O() with ϕ(α 1 ) = α 2 (cf. [29, Thm.2.4] or Remark 7.4). Thus, the assertion of the corollary is essentially equivalent to the proposition (see [1] page 111). Note that for general HKs we don t know which values of 2m can be realized. As a further trivial consequence, one finds that the set of those P Q such that P 0 is dense in Q. One can now go on and ask for those P Q such that P has higher rank. Those with maximal rank, i.e. rk(p ) = rk() 2, are called exceptional. An equivalent definition is Definition 4.9 A period point P Q is exceptional if P R is defined over Q, i.e. P Q P( Q(i) ).